How would I go about establishing a new offshore bank?

It depends which country you would choose to setup an off-shore bank. The main requirements would be to obtain a banking license from the central (or reserve) bank of that country. Setting up an off-shore bank typically means you are giving your clients some form of a tax break, so be very cognizant of the tax breaks the government would be providing for all banks who wish to setup in their region.

There are many small Island nations, etc. that you should research them. I've started maintaining an up-to-date list of all the central banks in the world, specifically look at Oceania section. Its not complete yet, but would be very soon.

Banks & Banking: Central Banks: List of Central & Reserve Banks of the World